find common difference of Ap whose first term is 5 and the sum of its four terms is half the sum of next four terms.
Answers
AnswEr :
- First term of AP is 5.
- Sum of first 4 terms is half of the sum of next four terms.
- Find the Common Difference.
Let the Common Difference be d.
⋆ Nth Term = { a + ( n - 1 )d }
• According to the Question Now :
⇒ Sum of First 4 Term = 1 /2 × ( Sum of Next Four Terms from First )
⇒ (a₁ + a₂ + a₃ + a₄) × 2 = (a₅ + a₆ + a₇ + a₈)
⇒ { (a) + (a + d) + (a + 2d) + (a + 3d) } × 2 = (a + 4d) + (a + 5d) + (a + 6d) + (a + 7d)
⇒ ( 4a + 6d ) × 2 = ( 4a + 22d )
⇒ 8a + 12d = 4a + 22d
⇒ 8a - 4a = 22d - 12d
⇒ 4a = 10d
⇒ 4 × 5 = 10d
⇒ 20 = 10d
- Dividing Both term by 10
⇒ d = 2
∴ Therefore, Common Difference will be 2.
Step-by-step explanation:
Given,
a1 = 5 (first term)
sum of first 4 terms = sum of 1/2 of next four terms
i.e a1+a2+a3+a4 = 1/2(a5+a6+a7+a8)
we know that an = a+(n-1)d
a+(a+d)+(a+2d)+(a+3d) = 1/2(a+4d)+(a+5d)+(a+6d)+(a+7d)
where "d" is the common diff...
Substituting 5 in places of "a" we get,
5+(5+d)+(5+2d)+(5+3d)= 1/2(5+4d)+(5+5d)+(5+6d)+(5+7d)
20+6d = 1/2(20+22d)
20+6d = 10+11d
5d=10
d=2